Adding Fractions Without Using the LCD

Date: 11/18/1999 at 09:48:12
From: Steve Russell
Subject: I found a new math formula
Once upon a time I was doing my math homework and I saw a problem. For
instance, say the problem was 5/6 + 7/12. Well I didn't want to find
the least common denominator, so I figured out a way to do it without
having to find the LCD. It takes a little more reducing but it works.
First you cross-multiply so you do 6 x 7 = 42 and 12 x 5 = 60. Then
you add them together and it equals 102. Then you multiply the
denominators 12 x 6 = 72 so you have 60/72. Then you reduce. It works
out - if you try doing it the other way it equals 5/6, and if you do
it my way it equals 5/6.
Will you please tell me if I found a new idea or not?
Thanks for your time.

Date: 11/18/1999 at 15:39:37
From: Doctor Rick
Subject: Re: I found a new math formula
Hi, Steve.
Your method works. It's not new, but it's a good observation. When you
add or subtract fractions, you need to express both fractions with the
same denominator. It doesn't have to be the *least* common
denominator; *any* common denominator will do. The product of the two
denominators works fine, as you have discovered; it is a multiple of
both denominators, and it's easy to find.
As you have noted, you still have work to do to reduce the result you
get. This work is the same as what you would do to find the least
common denominator. One way to find the LCD is to multiply the two
numbers and then divide by any factors that the two numbers had in
common: 6 * 12 = 72, but since both 6 and 12 have a factor of 6, you
divide 72 by 6 to get 12. When you simplified 102/72, you did the same
thing: you found that the two numbers are both divisible by 6, so you
divided through by 6 to get 17/12.
Often the LCD method is easier in the long run, because you have
smaller numbers to factor at the end if the answer needs to be
simplified. But I will sometimes use your method and put off the
factoring until the end.
It's good to be aware that there is more than one method to solve a
problem, so I hope you'll keep looking for methods other than what
you're taught.
- Doctor Rick, The Math Forum
http://mathforum.org/dr.math/

Date: 11/18/1999 at 16:56:47
From: Doctor Ian
Subject: Re: I found a new math formula
Hi Steve,
First, let me say that I admire your attitude, not wanting to do
things by the book all the time! That's the kind of attitude that
leads to discovery, and even when it doesn't, people with that
attitude seem to have a lot more fun.
Can I suggest that you go to the library and get a book called _Surely
You're Joking, Mr. Feynman_? It's the autobiography of a guy that I
think you'd really like, Richard Feynman. He was a great physicist
(winner of the Nobel Prize) and a great mathematician, but mostly he
was just a terrifically interesting guy, who practically _never_ did
anything by the book. As a result, I think he probably had more fun
than any three other guys I can think of.
Don't worry - it's an easy read, and to be honest, if I could go back
in a time machine and make one change to my life, it would be to give
this book to myself when I was your age. I didn't find out about it
until I was almost 30!
Anyway, on to your new method for adding fractions...
If you write down your method using letters instead of numbers, this
is what you get:
a c
- + - = ? original problem
b d
ad + bc cross-multiply and add
ad + bc
------- multiply denominators, and divide
bd
What's interesting is that what you're doing _is_ finding a common
denominator - it just isn't guaranteed to be the smallest one.
Here's another way of doing the same thing:
a c
- + - = ? original problem
b d
a(d) c(b)
---- + ---- multiply each term by n/n = 1
b(d) d(b)
ad + bc
------- simplify
bd
I think it's only because you're using numbers instead of letters that
you didn't see what was going on. Instead of fooling around dividing
the denominators up front, you're postponing all the division until
the final simplification step.
Anyway, the bad news is that you haven't really found a new way to add
fractions - although what really counts is that it was new to you, and
that no one told you how to do it.
The good news is that the method you're using is much easier than
finding the lowest common denominator all the time. (But you don't
need me to tell you that, do you?) I use your method instead of the
one they teach in the books, and I suspect that a lot of
mathematicians who have taken the time to play around a little do the
same.
Thanks for an interesting question, Steve. Check out the Feynman book,
and be sure to write back if you make any more discoveries, or if you
just have a question.
- Doctor Ian, The Math Forum
http://mathforum.org/dr.math/